Let W be the subspace of M_2x2 (R) consisting of the symmetric 2x2 matrices. The set

S= { , }

generates W. Find a subset of S that is a basis for W.

(I'm not sure what all the (0)'s are next to my matrices but they aren't supposed to be there its only supposed to be the 5 matrices)

I believe it was the "\\" just before \end{pmatrix} or the unecessary [ and \] that caused the (0)s. I have removed them here.

The simples thing to do, I think, is just try to choose matrices from the 5 given as generators that are independent.

For example,
is not a multiple of
so those two are independent.

Now, can
be written as a linear combination of the first two? That is, can you find numbers, a and b, such that
[tex]
\begin{pmatrix}
2 & 1 \\
1 & 9
\end{pmatrix}= a\begin{pmatrix}
1 & 2 \\
2 & 3
\end{pmatrix} + b\begin{pmatrix}
1 & -2 \\
-2 & 4
\end{pmatrix}[/MATh]?
If so, they they are dependent and you can discard the third matrix. If not, is the fourth matrix independent of the first three?